Quadratic Equations and Expressions
1.0 Definition
1.0 Definition
1. Polynomial: A function $f$ defined by $$f(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_1}x + {a_0}$$ where ${a_0},{a_1},{a_2},...,{a_n} \in R$ is called a polynomial of degree $n$ with real coefficients $({a_n} \ne 0,n \in W)$.
2. Linear equation: An equation of the form $$ax + b = 0\quad ...(1)$$ where $a,b \in R$ and $a \ne 0$, is a linear equation. Equation $(1)$ has a unique root equal to $ - \frac{b}{a}$.
3. Quadratic polynomial and Quadratic equation: A polynomial of degree $2$ is known as quadratic polynomial. Any equation $f(x) = 0$, where $f$ is a quadratic polynomial, is called a quadratic equation.
An equation of the form $$a{x^2} + bx + c = 0\quad ...(2)$$ where $a \ne 0$ and $a$, $b$, $c$ are real numbers, is called a quadratic equation. $a$, $b$, $c$ are called the coefficients of the quadratic equation.
4. Equation and Identity: Identity is a statement which is true for all values of variable whereas equations are the statements true for some or no values of the variable.
If $a{x^2} + bx + c = 0$ is satisfied by three distinct values of $x$, then it is an identity.
For example:
(a) ${(x + 2)^2} = {x^2} + 4x + 4$ is an identity.
(b) ${(x + 2)^2} = {x^2} + 4x + 3$ is an equation having no root.
(c) ${(x + 2)^2} = {x^2} + 5x + 3$ is an equation having $1$ as its root.
A quadratic equation has exactly two roots which may be real (equal or unequal) or imaginary.
$a{x^2} + bx + c = 0$ is:
| Type | Condition | Number of roots |
| a quadratic equation if | $a \ne 0$ | Two roots |
| a linear equation if | $a = 0,b \ne 0$ | One root |
| a contradiction if | $a = b = 0,c \ne 0$ | No root |
| an identity if | $a = b = c = 0$ | Infinite roots |
